Recall the product space of the Michael line and the space of the irrational numbers. Even though the first factor is a normal space (in fact a paracompact space) and the second factor is a metric space, their product space is not normal. This is one of the classic examples demonstrating that normality is not well behaved with respect to product space. This post presents an even more striking result, i.e., for any non-discrete normal space , there exists another normal space such that is not normal. The example of the non-normal product of the Michael line and the irrationals is not some isolated example. Rather it is part of a wide spread phenomenon. This result guarantees that no matter how nice a space is, a counter part can always be found that the product of the two spaces is not normal. This result is known as Morita’s first conjecture and was proved by Atsuji and Rudin. The solution is based on a generalization of Dowker’s theorem and a construction done by Rudin. This post demonstrates how the solution is put together.
All spaces under consideration are Hausdorff.
Morita’s First Conjecture
In 1976, K. Morita posed the following conjecture.
Morita’s First Conjecture
If is a normal space such that is a normal space for every normal space , then is a discrete space.
The proof given in this post is for proving the contrapositive of the above statement.
Morita’s First Conjecture
If is a non-discrete normal space, then there exists some normal space such that is not a normal space.
Though the two forms are logically equivalent, the contrapositive form seems to have a bigger punch. The contrapositive form gives an association. Each non-discrete normal space is paired with a normal space to form a non-normal product. Examples of such pairings are readily available. Michael line is paired with the space of the irrational numbers (as discussed above). The Sogenfrey line is paired with itself. The first uncountable ordinal is paired with (see here) or paired with the cube where with the usual topology (see here). There are plenty of other individual examples that can be cited. In this post, we focus on a constructive proof of finding such a pairing.
Since the conjecture had been affirmed positively, it should no longer be called a conjecture. Calling it Morita’s first theorem is not appropriate since there are other results that are identified with Morita. In this discussion, we continue to call it a conjecture. Just know that it had been proven.
Next, we examine Dowker’s theorem, which characterizes normal countably paracompact spaces. The following is the statement.
Theorem 1 (Dowker’s Theorem)
Let be a normal space. The following conditions are equivalent.
- The space is countably paracompact.
- Every countable open cover of has a point-finite open refinement.
- If is an open cover of , there exists an open refinement such that for each .
- The product space is normal for any compact metric space .
- The product space is normal where is the closed unit interval with the usual Euclidean topology.
- For each sequence of closed subsets of such that and , there exist open sets such that for each such that .
The theorem is discussed here and proved here. Any normal space that violates any one of the conditions in the theorem is said to be a Dowker space. One such space was constructed by Rudin in 1971 . Any Dowker space would be one factor in a non-normal product space with the other factor being a compact metric space. Actually much more can be said.
The Dowker space constructed by Rudin is the solution of Morita’s conjecture for a large number of spaces. At minimum, the product of any infinite compact metric space and the Dowker space is not normal as indicated by Dowker’s theorem. Any nontrivial convergent sequence plus the limit point is a compact metric space since it is homeomorphic to (as a subspace of the real line). Thus Rudin’s Dowker space has non-normal product with . Furthermore, the product of Rudin’s Dowker space and any space containing a copy of is not normal.
Spaces that contain a copy of extend far beyond the compact metric spaces. Spaces that have lots of convergent sequences include first countable spaces, Frechet spaces and many sequential spaces (see here for an introduction for these spaces). Thus any Dowker space is an answer to Morita’s first conjecture for the non-discrete members of these classes of spaces. Actually, the range for the solution is wider than these spaces. It turns out that any space that has a countable non-discrete subspace would have a non-normal product with a Dowker space. These would include all the classes mentioned above (first countable, Frechet, sequential) as well as countably tight spaces and more.
Therefore, any Dowker space, a normal space that is not countably paracompact, is severely lacking in ability in forming normal product with another space. In order to obtain a complete solution to Morita’s first conjecture, we would need a generalized Dowker’s theorem.
The key is to come up with a generalized Dowker’s theorem, a theorem like Theorem 1 above, except that it is for arbitrary infinite cardinality. Then a -Dowker space is a space that violates one condition in the theorem. That space would be a candidate for the solution of Morita’s first conjecture. Note that Theorem 1 is for the infinite countable cardinal only. Before stating the theorem, let’s gather all the notions that will go into the theorem.
Let be a space. Let be an open cover of the space . The open cover is said to be shrinkable if there is an open cover such that for each . When this is the case, the open cover is said to be a shrinking of . If an open cover is shrinkable, we also say that the open cover can be shrunk (or has a shrinking).
Let be a cardinal. The space is said to be a -shrinking space if every open cover of cardinality of the space is shinkable. The space is a shrinking space if it is a -shrinking space for every cardinal .
When a family of sets are indexed by ordinals, the notion of an increasing or decreasing family of sets is possible. For example, the family of subsets of the space is said to be increasing if whenever . In other words, for an increasing family, the sets are getting larger whenever the index becomes larger. A decreasing family of sets is defined in the reverse way. These two notions are important for some shrinking properties discussed here – e.g. using an open cover that is increasing or using a family of closed sets that is decreasing.
In the previous discussion on shrinking spaces, two other shrinking properties are discussed – property and property . A space is said to have property if every increasing open cover of cardinality for the space is shrinkable. A space is said to have property if every increasing open cover of cardinality for the space has a shrinking that is increasing. See this previous post for a discussion on property and property .
An Attempt for a Generalized Dowker’s Theorem
Let be an infinite cardinal. The space is said to be a -paracompact space if every open cover of with has a locally finite open refinement. Thus a space is paracompact if it is -paracompact for every infinite cardinal . Of course, an -paracompact space is a countably paracompact space.
For any infinite , let be a discrete space of size . Let be a point not in . Define the space as follows. The subspace is discrete as before. The open neighborhoods at are of the form where and . In other words, any open set containing contains all but less than many discrete points.
Another concept that is needed is the cardinal function called minimal tightness. Let be any space. Define the minimal tightness as the least infinite cardinal such that there is a non-discrete subspace of of cardinality . If is a discrete space, then let . For any non-discrete space , for some infinite . Note that for the space defined above would have . For any space , if and only if has a countable non-discrete subspace.
The following theorem can be called a -Dowker’s Theorem.
Let be a normal space. Let be an infinite cardinal. Consider the following conditions.
- The space is a -paracompact space.
- The space is a -shrinking space.
- For each open cover of , there exists an open cover such that for each .
- For each increasing open cover of , there exists an open cover such that for each .
- For each increasing open cover of , there exists an increasing open cover such that for each .
The following diagram shows how these conditions are related.
In addition to Diagram 1, we have the relations and .
At first glance, Diagram 1 might give the impression that the conditions in the theorem form a loop. It turns out the strongest property is -paracompactness (condition 1). Since condition 2 does not imply condition 5, condition 2 does not imply condition 1. Thus the conditions do not form a loop.
The implications and are immediate. The following implications are established in this previous post.
The remaining implications to be shown are and .
Proof of Theorem 2
Let be an increasing open cover of . By -paracompactness, let be a locally finite open refinement of . For each , define as follows:
Then is still a locally finite refinement of . Since the space is normal, any locally finite open cover is shrinkable. Let be a shrinking of . The open cover is also locally finite. For each , let . Then is an increasing open cover of . Note that
since is locally finite and thus closure preserving. Since is increasing, for all . This means that for all .
Since condition 3 is equivalent to condition 4, we show . Suppose that is normal where is a space such that . Let be a non-discrete subset of . Let be a point such that for all and such that is a limit point of (this means that every open set containing contains some ). Let be a decreasing family of closed subsets of such that . Define and as follows:
The sets and are clearly disjoint. The set is clearly a closed subset of . To show that is closed, let . Two cases to consider: or where is the first closed set in the family .
The first case . Let be least such that . Then for all since . In the space , any subset of cardinality is a closed set. Let , which is open containing . Let be open such that and . Then and misses points of .
Now consider the second case . Let be open such that and misses . Then is an open set containing such that misses . Thus is a closed subset of .
Since is normal, choose open such that and . For each , define as follows:
Note that each is open in and that for each . We claim that . Let . The point is in . Thus . Choose an open set such that and . Since , there is some such that . Since , . Thus . This establishes the claim that .
Analogous to the Dowker space, a -Dowker space is a normal space that violates one condition in Theorem 2. Since the seven conditions listed in Theorem 7 are not all equivalent, which condition to use? Condition 1 is the strongest condition since it implies all the other condition. At the lower left corner of Diagram 1 is condition 3, which follows from every other condition. Thus condition 3 (or 4) is the weakest property. An appropriate definition of a -Dowker space is through negating condition 3 or condition 4. Thus, given an infinite cardinal , a -Dowker space is a normal space that satisfies the following condition:
There exists a decreasing family of closed subsets of with such that for every family of open subsets of with for each , .
The definition of -Dowker space is through negating condition 4. Of course, negating condition 3 would give an equivalent definition.
When is the countably infinite cardinal , a -Dowker space is simply the ordinary Dowker space constructed by M. E. Rudin . Rudin generalized the construction of the ordinary Dowker space to obtain a -Dowker space for every infinite cardinal . The space that Rudin constructed in  would be a normal space such that condition 4 of Theorem 2 is violated. This means that the space would violate condition 7 in Theorem 2. Thus is not normal for every space with .
Here’s the solution of Morita’s first conjecture. Let be a normal and non-discrete space. Determine the least cardinality of a non-discrete subspace of . Obtain the -Dowker space as in . Then is not normal according to the preceding paragraph.
Answering Morita’s first conjecture is a two-step approach. First, figure out what a generalized Dowker’s theorem should be. Then a -Dowker space is one that violates an appropriate condition in the generalized Dowker’s theorem. By violating the right condition in the theorem, we have a way to obtain non-normal product space needed in the answer. The second step is of course the proof of the existence of a space that violates the condition in the generalized Dowker’s theorem.
Figuring out the form of the generalized Dowker’s theorem took some work. It is more than just changing the countable infinite cardinal in Dowker’s theorem (Theorem 1 above) to an arbitrary infinite cardinal. This is because the conditions in Theorem 1 are unequal when the cardinality is changed to an uncountable one.
We take the cue from Rudin’s chapter on Dowker spaces . In the last page of that chapter, Rudin pointed out the conditions that should go into a generalized Dowker’s theorem. However, the explanation of the relationship among the conditions is not clear. The previous post and this post are an attempt to sort out the conditions and fill in as much details as possible.
Rudin’s chapter did have the right condition for defining -Dowker space. It seems that prior to the writing of that chapter, there was some confusion on how to define a -Dowker space, i.e. a condition in the theorem the violation of which would give a -Dowker space. If the condition used is a stronger property, the violation may not yield enough information to get non-normal products. According to Diagram 1, condition 3 in Theorem 2 is the right one to use since it is the weakest condition and is down streamed from the conditions about normal product space. So the violation of condition 3 would answer Morita’s first conjecture.
We do not discuss the other step in the solution in any details. Any interested reader can review Rudin’s construction in  and . The -Dowker space is an appropriate subspace of a product space with the box topology.
One interesting observation about the ordinary Dowker space (the one that violates a condition in Theorem 1) is that the product of any Dowker space and any space with a countable non-discrete subspace is not normal. This shows that Dowker space is badly non-productive with respect to normality. This fact is actually not obvious in the usual formulation of Dowker’s theorem (Theorem 1 above). What makes this more obvious in the direction in Theorem 2. For the countably infinite case, is essentially this: If is normal where has a countable non-discrete subspace, then is not a Dowker space. Thus if the goal is to find a non-normal product space, a Dowker space should be one space to check.
In the course of working on the contents in this post and the previous post, there are some questions that we do not know how to answer and have not spent time to verify one way or the other. Possibly there are some loose ends to tie. They for the most parts are not open questions, but they should be interesting questions to consider.
For the -Dowker’s theorem (Theorem 2), one natural question is on the relative strengths of the conditions. It will be interesting to find out the implications not shown in Diagram 1. For example, for the three shrinking properties (conditions 2, 3 and 5), it is straightforward from definition that and . The example of (the first uncountable ordinal) shows that and hence . What about ? In , Beslagic and Rudin showed that using . A natural question would be: can there be ZFC example? Perhaps searching on more recent papers can yield some answers.
Another question is ? The answer is no with the example being a Navy space – Example 7.6 in p. 194 . The other two directions that have not been accounted for are: and ? We do not know the answer.
Another small question that we come across is about (the first uncountable ordinal). This is an example for showing . Thus condition 6 is false. Thus is not normal. Here is simply the one-point Lindelofication of a discrete space of cardinality . The question is: is condition 7 true for ? The product of and (a space with minimal tightness ) is not normal. Is there a normal where is another space with minimal tightness ?
Dowker’s theorem and -Dowker’s theorem show that finding a normal space that is not shrinking is not a simple matter. To find a normal space that is not countably shrinking took 20 years (1951 to 1971). For any uncountable , the -Dowker space that is based on the same construction of an ordinary Dowker space is also a space that is not -shrinking. With an uncountable , is the -Dowker space countably shrinking? This is not obvious one way or the other just from the definition of -Dowker space. Perhaps there is something obvious and we have not connected the dots. Perhaps we need to go into the definition of the -Dowker space in  to show that it is countably shrinking. The motivation is that we tried to find a normal space that is countably shrinking but not -shrinking for some uncountable . It seems that the -Dowker space in  is the natural candidate.
- Morita K., Nagata J.,Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
- Rudin M. E., A Normal Space for which is not Normal, Fund. Math., 73, 179-486, 1971. (link)
- Rudin M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
- Rudin M. E., -Dowker Spaces, Czechoslovak Mathematical Journal, 28, No.2, 324-326, 1978. (link)
- Rudin M. E., Beslagic A.,Set-Theoretic Constructions of Non-Shrinking Open Covers, Topology Appl., 20, 167-177, 1985. (link)
- Yasui Y., On the Characterization of the -Property by the Normality of Product Spaces, Topology and its Applications, 15, 323-326, 1983. (abstract and paper)
- Yasui Y., Some Characterization of a -Property, TSUKUBA J. MATH., 10, No. 2, 243-247, 1986.